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An inverse geometry problem for a one-dimensional heat equation: advances with complex temperatures

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This paper presents an inverse problem in heat conduction, namely the determination of thicknesses of three materials of known heat capacities and thermal conductivities inside a rod of given length subjected to periodic heat flows from measurements of temperatures at both ends. The unknowns are, therefore, the positions of the two interior frontiers between the three materials. Classically, they can be obtained by minimizing the least-squares, non-linear criterion, between the measured and calculated temperatures. Nevertheless, we show that the global minimum providing the solution is close to three local minima that act as traps for a descent algorithm. After providing theoretical justification of the complex temperature method, a method based in this case on the periodicity of boundary fluxes, we suggest a new criterion allowing the global characterization or not of an a priori local minimizer to be tested. It is a criterion of topological nature based on the identification of a singularity.

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https://hal.univ-angers.fr/hal-03287227
Contributeur : Okina Université d'Angers <>
Soumis le : jeudi 15 juillet 2021 - 14:38:51
Dernière modification le : vendredi 16 juillet 2021 - 08:54:02

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Jean-Claude Jolly, Laetitia Perez, Laurent Autrique. An inverse geometry problem for a one-dimensional heat equation: advances with complex temperatures. Inverse Problems in Science and Engineering, Taylor & Francis, 2014, 22 (1), pp.63-83. ⟨10.1080/17415977.2013.827186⟩. ⟨hal-03287227⟩

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